Topological chaos of universal elementary cellular automata rule

Abstract

The dynamical behaviors of elementary cellular automata rule 110 are analyzed from the viewpoint of symbolic dynamics in the space of bi-infinite symbolic sequences. This paper conducts a rigorous analysis of the relationship between rules 110, 170 and 240 by applying blocking transformation and releasing transformation. Based on this result, the topological chaos of T 110 induced by rule 110 is evaluated; that is, T 9 110 and T 16 110 are topologically mixing and possess the positive topological entropies on their respective subsystems. Therefore, it is natural to argue that the intrinsic complexity of rule 110 is high according to the usual measure of complexity organized around the symbolic dynamics of stationary symbol sequences. Finally, it is worth mentioning that the method presented in this paper is also applicable to other blocking transformation equivalences therein.

Chaos emerged on the ‘edge of chaos’

“…In spite of the existence of a chaotic universal CA has not been confirmed, but this paper has found some chaotic tracks in the CA on 'the edge of chaos', and hence partially answered the question 'whether an example of chaotic universal CA exists' posted in [15]. [22] that f 9 110 and f 16 110 are chaotic in the sense of both Li-Yorke and Devaney on their respective invariant sets by using blocking and releasing transformations. However, the structures of the two invariant sets and the dynamics of the original f 110 have not been characterized.…”

“…However, the structures of the two invariant sets and the dynamics of the original f 110 have not been characterized. We expect that the two invariant sets derived from [22] should be subsets of 2 or 1 based on the present work, because f 3…”

“…Nevertheless, the 'edge of chaos' has been intensively studied for CA, yet it is still unknown as if an example of truly chaotic universal CA exists [15]. Previously, we proved that Rule 110 possesses a positive topological entropy, implying it is chaotic in the sense of Li-Yorke [22]. Of course, Li-Yorke chaos is weaker than Devaney chaos [21].…”

“…Since 2002, Chua et al have provided a nonlinear dynamics perspective to Wolfram's empirical observations from mathematical analysis, using concepts like the characteristic function, forward time-τ map, basin tree diagram and the Isle-of-Eden digraph [7][8][9][10]. Recently, the dynamical properties of Chua's periodic rules and the robust Bernoulli-shift rules with distinct parameters have also been investigated from the viewpoint of symbolic dynamics [2][3][4][5][6][22][23][24].…”

Infinite Number of Disjoint Chaotic Subsystems of Cellular Automaton Rule 106

“…Let [19] [20], for example, rule 180 possesses infinitely many generalized sub-shifts [20] [21]. This paper is devoted to an in-depth study of cellular automaton rule 106 in the framework of symbolic dynamics.…”